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Mirrors > Home > MPE Home > Th. List > Mathboxes > sepnsepolem2 | Structured version Visualization version GIF version |
Description: Open neighborhood and neighborhood is equivalent regarding disjointness. Lemma for sepnsepo 48050. Proof could be shortened by 1 step using ssdisjdr 47987. (Contributed by Zhi Wang, 1-Sep-2024.) |
Ref | Expression |
---|---|
sepnsepolem2.1 | β’ (π β π½ β Top) |
Ref | Expression |
---|---|
sepnsepolem2 | β’ (π β (βπ¦ β ((neiβπ½)βπ·)(π₯ β© π¦) = β β βπ¦ β π½ (π· β π¦ β§ (π₯ β© π¦) = β ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sepnsepolem2.1 | . 2 β’ (π β π½ β Top) | |
2 | id 22 | . . 3 β’ (π½ β Top β π½ β Top) | |
3 | sslin 4230 | . . . . 5 β’ (π§ β π¦ β (π₯ β© π§) β (π₯ β© π¦)) | |
4 | sseq0 4396 | . . . . . 6 β’ (((π₯ β© π§) β (π₯ β© π¦) β§ (π₯ β© π¦) = β ) β (π₯ β© π§) = β ) | |
5 | 4 | ex 411 | . . . . 5 β’ ((π₯ β© π§) β (π₯ β© π¦) β ((π₯ β© π¦) = β β (π₯ β© π§) = β )) |
6 | 3, 5 | syl 17 | . . . 4 β’ (π§ β π¦ β ((π₯ β© π¦) = β β (π₯ β© π§) = β )) |
7 | 6 | adantl 480 | . . 3 β’ ((π½ β Top β§ π§ β π¦) β ((π₯ β© π¦) = β β (π₯ β© π§) = β )) |
8 | ineq2 4201 | . . . . 5 β’ (π¦ = π§ β (π₯ β© π¦) = (π₯ β© π§)) | |
9 | 8 | eqeq1d 2727 | . . . 4 β’ (π¦ = π§ β ((π₯ β© π¦) = β β (π₯ β© π§) = β )) |
10 | 9 | adantl 480 | . . 3 β’ ((π½ β Top β§ π¦ = π§) β ((π₯ β© π¦) = β β (π₯ β© π§) = β )) |
11 | 2, 7, 10 | opnneieqv 48037 | . 2 β’ (π½ β Top β (βπ¦ β ((neiβπ½)βπ·)(π₯ β© π¦) = β β βπ¦ β π½ (π· β π¦ β§ (π₯ β© π¦) = β ))) |
12 | 1, 11 | syl 17 | 1 β’ (π β (βπ¦ β ((neiβπ½)βπ·)(π₯ β© π¦) = β β βπ¦ β π½ (π· β π¦ β§ (π₯ β© π¦) = β ))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 394 = wceq 1533 β wcel 2098 βwrex 3060 β© cin 3940 β wss 3941 β c0 4319 βcfv 6543 Topctop 22808 neicnei 23014 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-iun 4994 df-br 5145 df-opab 5207 df-mpt 5228 df-id 5571 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-top 22809 df-nei 23015 |
This theorem is referenced by: sepnsepo 48050 |
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